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Gradients
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Foundational Skills
Identify positive or negative gradient from graph
\[ m > 0 \text{ or } m < 0 \text{?} \]
Determine whether a line has a positive or negative gradient.
Identify steeper line from two graphs
\[ \text{Which is steeper: A or B?} \]
Compare two lines and identify which has the steeper gradient.
Match gradient to line description
\[ \text{Up 2 for every 1 across} \rightarrow m = 2 \]
Find the gradient given a verbal description of rise and run.
Describe line from gradient
\[ m = 3 \rightarrow \text{description} \]
Describe the rise and run for a given gradient.
Identify gradient as rate of change
\[ m = 5 \text{ means…} \]
Interpret gradient as a rate of change in context.
Identify zero gradient
\[ \text{Horizontal line: } m = 0 \]
Recognise that horizontal lines have gradient zero.
Identify undefined gradient
\[ \text{Vertical line: } m = \text{?} \]
Recognise that vertical lines have undefined gradient.
Calculating Gradient
Calculate gradient (positive)
\[ (1, 2) \text{ and } (3, 8) \rightarrow m = 3 \]
Calculate gradient from two points where the answer is positive.
Calculate gradient (negative)
\[ (1, 6) \text{ and } (4, 0) \rightarrow m = -2 \]
Calculate gradient from two points where the answer is negative.
Calculate gradient (fractional)
\[ (0, 1) \text{ and } (4, 3) \rightarrow m = \frac{1}{2} \]
Calculate gradient where the answer is a fraction.
Calculate gradient (negative coords)
\[ (-2, -3) \text{ and } (1, 3) \rightarrow m \]
Calculate gradient with negative coordinates.
Calculate gradient from table
\[ \text{x: 1, 3, 5; y: 2, 6, 10} \]
Calculate gradient from a table of x and y values.
Find missing coordinate
\[ m = 2, (1, 3), (4, y) \rightarrow y = ? \]
Find a missing coordinate using the gradient formula.
Find missing coordinate (harder)
\[ m = -\frac{1}{2}, (2, 5), (x, 3) \rightarrow x \]
Find missing x-coordinate with fractional gradient.
Gradient from Graphs
Read gradient from graph (positive)
\[ \text{Graph} \rightarrow m = 2 \]
Read positive integer gradient from a coordinate grid.
Read gradient from graph (negative)
\[ \text{Graph} \rightarrow m = -3 \]
Read negative integer gradient from a coordinate grid.
Read gradient from graph (unit fraction)
\[ \text{Graph} \rightarrow m = \frac{1}{2} \]
Read unit fraction gradient from a coordinate grid.
Read gradient from graph (other fraction)
\[ \text{Graph} \rightarrow m = \frac{2}{3} \]
Read non-unit fraction gradient from a coordinate grid.
Find gradient between marked points
\[ \text{Points A and B} \rightarrow m \]
Calculate gradient between two marked points on a graph.
Gradient from Equation
Gradient from y = mx + c (positive m)
\[ y = 3x + 2 \rightarrow m = 3 \]
Identify positive gradient from y = mx + c form.
Gradient from y = mx + c (negative m)
\[ y = -2x + 5 \rightarrow m = -2 \]
Identify negative gradient from y = mx + c form.
Gradient from y = mx + c (fractional m)
\[ y = \frac{1}{2}x – 3 \rightarrow m = \frac{1}{2} \]
Identify fractional gradient from equation.
Gradient from y = mx (no c term)
\[ y = 4x \rightarrow m = 4 \]
Identify gradient when there is no constant term.
Gradient from y = c (horizontal line)
\[ y = 5 \rightarrow m = 0 \]
Recognise that y = c has gradient zero.
Gradient from rearranged equation
\[ y – 2x = 5 \rightarrow m = 2 \]
Find gradient when equation needs rearranging.
Gradient from ax + by = c form
\[ 2x + 3y = 6 \rightarrow m = -\frac{2}{3} \]
Find gradient by rearranging ax + by = c.
Gradient from 2y = mx + c form
\[ 2y = 6x + 4 \rightarrow m = 3 \]
Find gradient when y has a coefficient other than 1.
Comparing Gradients
Compare gradients of two equations
\[ y = 2x + 1 \text{ vs } y = 5x – 3 \]
Compare gradients of two lines given their equations.
Identify parallel lines from gradients
\[ \text{Parallel? Same } m \]
Determine if two lines are parallel by comparing gradients.
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